Properties

Label 203490.ee
Number of curves $8$
Conductor $203490$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ee1")
 
E.isogeny_class()
 

Elliptic curves in class 203490.ee

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
203490.ee1 203490g8 \([1, -1, 1, -26135392757, -1625317462284019]\) \(2708215857449597952771459256806409/1815677562935478375000000000\) \(1323628943379963735375000000000\) \([6]\) \(477757440\) \(4.7183\)  
203490.ee2 203490g5 \([1, -1, 1, -26131186022, -1625867129640931]\) \(2706908330196708836642873424493849/816939805815000\) \(595549118439135000\) \([2]\) \(159252480\) \(4.1690\)  
203490.ee3 203490g6 \([1, -1, 1, -1956594677, -14632320905971]\) \(1136315122909965387044499819529/530704359775758422016000000\) \(386883478276527889649664000000\) \([2, 6]\) \(238878720\) \(4.3717\)  
203490.ee4 203490g2 \([1, -1, 1, -1633199342, -25403860641859]\) \(660866552951225193140994678169/363054521201227329600\) \(264666745955694723278400\) \([2, 2]\) \(79626240\) \(3.8224\)  
203490.ee5 203490g4 \([1, -1, 1, -1624013942, -25703738232739]\) \(-649778658927959232413187423769/15498405515425377751317720\) \(-11298337620745100380710617880\) \([2]\) \(159252480\) \(4.1690\)  
203490.ee6 203490g3 \([1, -1, 1, -1001079797, 12034570571021]\) \(152195662006675487969752714249/2254051004206282702848000\) \(1643203182066380090376192000\) \([6]\) \(119439360\) \(4.0252\)  
203490.ee7 203490g1 \([1, -1, 1, -102649262, -392223454531]\) \(164083032511008797673646489/3779535863669623787520\) \(2755281644615155741102080\) \([2]\) \(39813120\) \(3.4759\) \(\Gamma_0(N)\)-optimal
203490.ee8 203490g7 \([1, -1, 1, 6933965323, -110639700233971]\) \(50575615882668425252678113940471/36522079745400816582633408000\) \(-26624596134397195288739754432000\) \([6]\) \(477757440\) \(4.7183\)  

Rank

sage: E.rank()
 

The elliptic curves in class 203490.ee have rank \(0\).

Complex multiplication

The elliptic curves in class 203490.ee do not have complex multiplication.

Modular form 203490.2.a.ee

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + q^{5} + q^{7} + q^{8} + q^{10} + 2 q^{13} + q^{14} + q^{16} - q^{17} + q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 6 & 12 & 4 & 12 & 4 \\ 3 & 1 & 6 & 2 & 4 & 12 & 4 & 12 \\ 2 & 6 & 1 & 3 & 6 & 2 & 6 & 2 \\ 6 & 2 & 3 & 1 & 2 & 6 & 2 & 6 \\ 12 & 4 & 6 & 2 & 1 & 12 & 4 & 3 \\ 4 & 12 & 2 & 6 & 12 & 1 & 3 & 4 \\ 12 & 4 & 6 & 2 & 4 & 3 & 1 & 12 \\ 4 & 12 & 2 & 6 & 3 & 4 & 12 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.